Theorem breq12 4034

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 4032	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 4033	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   class class class wbr 4029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030

This theorem is referenced by:  breq12i  4038  breq12d  4042  breqan12d  4045  posng  4731  isopolem  5865  poxp  6285  rbropapd  6295  ecopover  6687  ecopoverg  6690  ltdcnq  7457  recexpr  7698  ltresr  7899  reapval  8595  ltxr  9841  xrltnr  9845  xrltnsym  9859  xrlttr  9861  xrltso  9862  xrlttri3  9863  xposdif  9948  f1olecpbl  12896  exmidsbthrlem  15512